Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

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JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

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Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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September 25, 2022 at 12:01am

#1038115

O Fortuna

Today's article is fairly long, but worth it if you have the time.

Fortune’s Wheel
For many in Western history, games of chance represented a portal of possibility, not a heresy to be demonized or a statistical probability to be managed.

In the American mythology of success, labor is the only path to prosperity.

Ha!

The affluent can cleanse their cash by claiming they worked hard for it; mastering fate and controlling outcomes bestow moral legitimacy on their earnings.

HA!

Many moralists throughout American history have affirmed that merit matches reward and that people get what they deserve, in this world and the next.

At the risk of repeating myself, ha.

This all ties in to another thing I featured over two years ago: "On Merit"

But a heresy against this faith in hard work has stubbornly survived nearly two centuries’ preaching of the virtues of America’s civil religion.

"Civil" religion, my lazy heretical ass. This nonsense about "hard work will get you ahead" can be directly traced to Calvin. The preacher, not the cartoon kid.

The lottery ticket, humble as it is, serves as a passport to a more fluid moral economy, where fate can be cruel or kind but is always arbitrary—where luck, as even Horatio Alger realized, matters more than pluck. And this culture of chance more closely resembles the world in which most people live than the one prescribed by the dominant mythology of success, which can aptly be called a culture of control.

Much has been said—including by me—about the problems with playing the lottery. But for many, it's the only thin straw they can grasp at to maybe escape their life at the bottom of the social hierarchy, at least for a while (more on that in another entry).

But it does suggest that gambling is about more than mere money. Modern games of chance reenact ancient rituals of divination—casting lots, throwing pebbles, bones, shells, or dice—designed to provide glimpses of the sacred and to conjure luck or its spiritual equivalent, grace. Rather than the static and timeless cosmic order of orthodox monotheism, the sense of the sacred sought by diviners was a pluralist plentitude, symbolized in Western tradition by inconstant Fortuna and by similar figures in American Indian and African traditions.

Yeah, that might be a stretch.

The article goes on to describe the history of Fortuna, which is quite fascinating in itself.

But Fortune did not fit well with Christian ideas of Providence. To early Christians, the divine plan unfolded as mysteriously as the fluctuations of luck, but however remote the planner or apparently perverse his decrees, his purpose was ultimately benign.

To me, as an outsider, "divine will" is indistinguishable from random chance working through the laws of physics.

Beginning in the fifteenth century, Protestant reformers assaulted these rituals as part of a broader war on the medieval culture of chance. Taking their cues from John Calvin, theologians disparaged Fortuna, deriding belief in her powers as a pagan excrescence on the Church.

See? I told you we'd get to Calvin.

Only by God’s “secret plan,” Calvin wrote, do “some distinguish themselves, while others remain contemptible.”

In the millennia of lies foisted on an unsuspecting public, this one is in the running for the worst in terms of the effect it had on society.

As early as 1653, when dissenting sects proliferated amid the English Civil War, a female sectarian confessed that she could not stand to see her neighbors prosper, as it meant they had prayed more than she had.

That, for example.

Despite occasional revolts, faith in Fortune endured in a variety of ways, even in the language that people used to describe their circumstances. The word “happiness” has long been linguistically dependent on chance. The thirteenth-century English substantive “hap” derives from the old Norse “happ,” meaning “chance” or “good luck.” The verb “happen” and the adverb “haply” (by chance) emerged from this root in the fourteenth century, as did “happy,” which originally meant “prosperous” and by the sixteenth century had acquired the connotation of contentment.

I'm just leaving this there as further support for my continued ragging on the incessant popular harping about "happiness."

The Protestant war on Fortune, declared by John Calvin centuries earlier, also allied itself with Newtonian science, whose practitioners were less interested in denying chance than in containing it. By the end of the eighteenth century, Jacob Bernoulli, Adolphe Quetelet, and other statisticians had developed modern probability theory, reducing chance to a predictable outlier or a standard deviation. Both statisticians and devotees of “rational religion” hastened the shift from a respectful and even fearful Renaissance vision of Fortune as a goddess to a modern, more confident understanding of chance as a condition to be managed.

Yeah, I don't fully buy that bit. Newton envisioned a purely deterministic universe, and probability theory is kind of not that.

As Wall Street sanitized speculation as “investment,” and the mere manipulation of money through complex financial instruments became a path to self-made manhood, gambling continued to be stigmatized by society.

This I don't buy either. Sure, you can enter the stock market speculatively, risking capital on short-term investments. But there's one very important difference between investing and speculation: Insofar as investing can be considered subject to the whims of chance, historically, long-term investments are the equivalent of owning the casino, giving the investor the house edge.

Mason Locke Weems, whose early biography of George Washington introduced American readers to the story of the cherry tree, condemned gambling’s effect on the “social body.”

As the cherry tree story is pretty firmly in the realm of mythology, I wouldn't trust anything else that guy wrote, however well-intentioned.

The emergence of gambling as mass entertainment can be traced to several sources, including the willingness of cash-strapped state governments to substitute gambling revenues for taxation and the decline of job security among gamblers themselves—in a contingent labor market, where one can be dismissed for reasons having nothing to do with one’s performance, the disjunction between merit and reward is more painfully apparent than ever. If hard work gets you nowhere fast, why not have a fling with Fortune?

They sold us on a lottery here in Virginia, back in the 80s, saying the profits would go to transportation and education. As far as I know, they did—but the legislature ended up cutting off more traditional sources of funding to those areas, leaving us worse off than before.

The willingness to relinquish control over outcomes—to play—promoted the insouciance toward money that lies at the core of the culture of chance. This outlook arose from an insight common among Fortuna’s children: the recognition that you don’t get what you deserve. You get what you get.

The article, and I, mentioned Newton up there. As I said, he saw a deterministic world, where calculations had fixed results and if you but had enough information, you could make any prediction accurately.

And that turns out not to be the case.

so at this hour
without delay
pluck the vibrating strings;
since Fate
strikes down the strong man,
everyone weep with me!
—O Fortuna

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